construction of monolithic all-glass optical...

CONSTRUCTION OF MONOLITHIC ALL-GLASS OPTICAL

CAVITIES FOR TRAPPING ULTRACOLD ATOMS

A Thesis

Submitted to the Faculty

in partial fulfillment of the requirements for the

degree of

Masters of Science

in

Physics

by

Jesse Evans

DARTMOUTH COLLEGE

Hanover, New Hampshire

March 23, 2017

Kevin C. Wright, Advisor

Alex J. Rimberg

Kristina A. Lynch

F. Jon Kull, Ph.D.Dean of Graduate Studies

Abstract

This thesis reports a successful testing of a procedure for the construction of a high-

finesse monolithic glass bowtie cavity using hydroxide catalysis bonding. The design

of the cavity is optimized for use in optical dipole trapping of ultracold atoms. The

all-glass design is especially suited for experiments that include rapidly changing mag-

netic field, such as those employing Feshbach resonances. These all-glass hydroxide

catalysis bonded cavities are ultra-high vacuum compatible and survive moderate

bake-out temperatures required to reach ultra low pressures.

We developed a set of alignment procedures and mechanical jigs for the purposes

of precisely aligning a symmetric bowtie cavity. The alignment procedures and jigs

guarantee a minimal degree of cavity twist and maximize overlap at the beam crossing

point. A prototype cavity constructed using these procedures was measured to have

a beam overlap within the tolerances necessary to generate an effective crossed beam

dipole trap.

ii

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Cooling and Trapping Li atoms in a MOT . . . . . . . . . . . 2

1.1.2 Crossed Dipole Beam Trap and Evaporation . . . . . . . . . . 6

1.2 Optical Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.1 Cavity Properties . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.2 Cavity Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.3 Cavity Matrices and q Parameter . . . . . . . . . . . . . . . . 17

1.2.4 Cavity Locking . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3 Experimental Design Constraints . . . . . . . . . . . . . . . . . . . . 19

1.3.1 Trapping Atoms in Cavities . . . . . . . . . . . . . . . . . . . 20

1.3.2 Atom-Cavity Physics . . . . . . . . . . . . . . . . . . . . . . . 23

1.4 Cavity Construction Techniques . . . . . . . . . . . . . . . . . . . . . 26

1.4.1 Standard Optomechanics . . . . . . . . . . . . . . . . . . . . . 27

1.4.2 Monolithic Techniques . . . . . . . . . . . . . . . . . . . . . . 27

2 Development Process and Methods 30

iii

2.1 Prior Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2 Hydroxide Catalysis Bonding Tests . . . . . . . . . . . . . . . . . . . 30

2.3 Bowtie Cavity Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.5 Pound Drever Hall Locking . . . . . . . . . . . . . . . . . . . . . . . 38

3 Results 46

3.1 Prototypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.1.1 Table-Mounted Linear Cavity 1 . . . . . . . . . . . . . . . . . 47

3.1.2 Monolithic Linear Cavity 1 . . . . . . . . . . . . . . . . . . . . 48

3.1.3 Monolithic Linear Cavity 2 . . . . . . . . . . . . . . . . . . . . 51

3.1.4 Table-Mounted Bowtie . . . . . . . . . . . . . . . . . . . . . . 53

3.1.5 Table-Mounted Linear Cavity 2 . . . . . . . . . . . . . . . . . 55

3.2 Monolithic Bowtie Cavity . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2.1 Jig Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2.2 Alignment Procedure . . . . . . . . . . . . . . . . . . . . . . . 60

3.2.3 Verification of the Cavity Alignment . . . . . . . . . . . . . . 63

4 Conclusions 66

A Part Numbers 68

B mode matching Procedure 70

References 73

Acknoledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

iv

Chapter 1

Introduction

The ultimate goal of this project is to design an optical cavity that allows for power

buildup in a crossed-beam dipole trap for lithium 6 and lithium 7, leading to evapo-

rative cooling of the atoms to the point of quantum degeneracy. In the grand scheme

of cold atom physics, quantum degenerate atom mixtures are the foundation upon

which more cutting-edge physics relies. In the long run, our group seeks to create

ring Bose-Einstein Condensates (BECs) and Fermi gases while exploring current top-

ics in ultracold physics such as 1D superfluid rings, the BEC-BCS crossover [15], the

Fulde–Ferrell-Larkin-Ovchinnikov state [13], and effective dynamic gauge fields [6].

Exploration of all these topics is impossible until we create a quantum degenerate

sample.

On the road to quantum degeneracy, the atomic mixture is first cooled in a

magneto-optical trap (MOT) inside an ultra-high vacuum (UHV) chamber. The

next step is to capture the mixture in a crossed-beam dipole trap consisting of two

red-detuned lasers. The intensity of the optical fields is adiabatically decreased, low-

ering the trapping potential and allowing more energetic atoms to escape and causing

1

1.1 Motivation Introduction

the captured atoms to re-thermalize at lower temperatures. A strong magnetic field

is applied to take advantage of the Feshbach resonance’s ability to tune atom-atom

interactions, giving us control over the thermalization rate of the ensemble. When

the atoms reach critical temperatures at a high phase space density, they become

quantum degenerate, either in the form of a BEC or a quantum degenerate Fermi

gas.

The scope of my work did not include cooling to degeneracy or exploring inter-

esting many-body physics states, but it is necessary to understand each step along

the way and to keep the end goals in mind when designing the dipole trap. Atypi-

cal design constraints have forced us to eschew conventional construction techniques

and develop techniques of our own to work within experimental limits. I will outline

both the constraints and the benefits reaped by adhering to them in the rest of this

chapter.

1.1 Motivation

1.1.1 Cooling and Trapping Li atoms in a MOT

Our group has successfully cooled both 7Li and 6Li to millikelvin temperatures in

a 3D MOT. I have included a picture of the MOT chamber and images of trapped

lithium 6 and 7 from our experiment in Fig. 1.1

A MOT consists of a quadrupole magnetic field and counter-propagating lasers

with frequencies close to a strong optical transition of the atomic species that will

be cooled. The lasers are red-detuned from the cooling transition, so that when the

atoms are moving towards one of the laser beams the Doppler shift induced by their

2


Figure 1.1: Lithium 6 and 7 MOT. Top Right: Trapped lithium 6. Bottom Left:Image of trapped lithium 7. Background Layer: The MOT experimental chamber

velocity causes the incident light to become resonant with the atoms, increasing the

chances of absorbing a photon from the beam. The atom experiences a momentum

kick of ~k per photon absorbed, decreasing its average kinetic energy and resulting

in “Doppler cooling”. The quadrupole magnetic field has a minimum value (zero) at

the center of the trap and an approximately linear positive field gradient in the local

neighborhood of the zero. The field gradient results in a Zeeman shift of the hyperfine

mf sublevels of the atoms. Therefore, as the atoms move away from the field center

their resonant frequencies shift closer to that of the detuned lasers, resulting in a

restoring force that pushes them back to the center. A MOT with 3 sets of counter-

propagating red-detuned lasers intersecting at right angles in a 3D quadrupole field,

can trap atoms atoms along all 3 spatial direction and cool them to temperatures near

the “Doppler limit” TD = ~Γ/2kB, where Γ is the linewidth of the transition, and kB

is Boltzmann’s constant. For lithium, this temperature is well below 1 millikelvin.

3


In our setup, we made use of a 2D+ MOT and a 3D MOT to cool both 6Li

and 7Li. In a UHV chamber we loaded both isotopes into the 2D+ MOT and applied

counter-propagating red-detuned beams along the x and y directions, confining atoms

along the z axis. We used a resonant “push beam” to propel the atoms along the

z axis, creating a cold atom stream into the 3D MOT chamber. We then trapped

the atoms from the stream in the center of the 3D MOT using 3 sets of orthogonal

counter-propagating beams in a quadrupole magnetic field.

Laser cooling of alkali metal atoms usually employs light that is near-resonant

with the 2S →2 P transitions, historically referred to as the Fraunhofer “D” lines.

The fine structure interaction splits the 2P electronic state into 2P1/2 and n2P3/2.

The transition from the ground state to the n2P3/2 (referred to as the D2 transition)

is the one typically used for laser cooling of alkali metals. All the alkali atoms have

nonzero nuclear spin, and the hyperfine interaction splits the 2S1/2 ground state in

two as well. Two laser frequencies are therefore required for laser cooling, because if

only one of the ground state manifolds is strongly coupled to the laser light, when

the atoms undergo spontaneous emission from the excited state back to the ground

state they may decay into the non-coupled manifold, and cease to interact with the

cooling laser. The (stronger) transition coupling the upper hyperfine ground state to

the excited state is conventionally called the “pump” transition, while the transition

to the lower hyperfine ground state is called the “repump”.

The specific transitions used for laser cooling of the lithium isotopes are shown in

Fig. 1.2. To cool 7Li, the pump and repump beams couple from the the F = 2 and

F = 1 ground state manifolds (respectively) to the 2P3/2 excited state. Similarly,

for 6Li the pump and repump beams couple from the F = 3/2 and F = 1/2 ground

4


Figure 1.2: Fine and hyperfine structure of the 2S → 2P transitions typically usedin laser cooling of 6Li (left) and 7Li (right).

2 2S1/2

2 2S1/2

2 1/2P1/2 2 1/2P1/2

2 1/2P3/22 1/2P3/2

F = 5/2

F = 3/2

F = 1/2

F = 1/2

F = 3/2

F = 1

F = 2

F = 1

F = 0

F = 2

F = 3

Lithium 6 Lithium 7

D2

D1

D1

D2

F = 3/2

F=1/2

F = 2

F = 1

Pump

Repump

Pump

Repump

state manifolds to the 2P3/2 excited states. In both lithium isotopes, the hyperfine

splitting of the 2P3/2 (D2) excited states is smaller than the transition line width,

and therefore unresolved.

Cooling and trapping the atoms in a MOT is the first step towards quantum

degeneracy and dictates the starting conditions for the atoms we intend to trap in a

crossed dipole beam setup. The starting atom numbers and temperatures for both

species depend on the efficiency of the MOT. Other experiments have achieved MOTs

with number densities of around 1010 cm−3 and temperatures below one mK [18], and

we expect to be in the same neighborhood with our setup.

5


1.1.2 Crossed Dipole Beam Trap and Evaporation

Once our lithium atoms are cooled and trapped in a MOT, we need to cool them

down to microkelvin temperatures to achieve high enough phase space density to

reach quantum degeneracy. One of the more common ways to reach these low tem-

peratures is to trap atoms in a crossed beam dipole trap and slowly lower the trapping

confinement in a process known as forced evaporative cooling.

Optical Dipole Trapping

When light is incident on an atom, it induces an oscillating electric dipole moment

on the atom given by

~p = α~E (1.1)

where α is the atomic polarizability and E is the electric field of the light wave. The

potential of the dipole moment is given by

U = − 1

2ε0cRe(α)I(r) (1.2)

where I have substituted working with the intensity of the light field instead of the

electric field. We can assume a Gaussian beam mode for the light field, whose spatial

intensity distribution is

I(r) = I0

(w0

w(z)

)2

e−r2/w(z)2 (1.3)

where w(z) is the beam diameter at z and w0 is the beam diameter at the waist, or

when the beam diameter is smallest. Note that the beam comes to a focus at the

waist, where the intensity decreases away from z = 0. The magnitude of the potential

is therefore highest along the axis of the beam and at the beam waist. The sign of

6


the potential depends on the sign of the atomic polarizability, which is given by

α = 6πε0c3 Γ/ω2

0

ω20 − ω2 − i(ω3/ω2

0)Γ(1.4)

for a given atomic transition at transition frequency ω0, where Γ is the damping factor

corresponding to the transition, given by

Γ =ω3

0

3πε0~c3| 〈e|g〉 |2 (1.5)

where 〈e| and 〈g| refer to ground (initial) and excited (final) states. Putting it all

together, we finally have the general form for the potential, given by

Udip = − 3πc2

2~ω30

(Γ

ω0 − ω+

Γ

ω0 + ω

)I(r) (1.6)

The difference between the resonance frequency ω0 and the laser frequency ω is gen-

erally referred to as the “detuning”. The rate of photon scattering can be derived

from the power loss, such that

Γsc =Pabs~ω

=1

~ε0cIm(α)I(r)

=3πc2

2~ω30

( ωω0

)3(

Γ

ω0 − ω+

Γ

ω0 + ω

)2

I(r)

(1.7)

Photon scattering involves the absorption and re-emission of a photon by an atom,

and contributes to heating. Because the scattering rate is inversely proportional to

the square of the detuning, whereas the trapping potential is inversely proportional to

first order, we design our trapping wavelengths to be far from resonance to minimize

7


incidental heating. We detune far to the red (below resonance frequency) to make our

trap high-field seeking, with a potential minimum at the focus of the beam. Note that

trapping confinement is strongest along the radial direction of the laser, and weaker

axially. To avoid the axial weakness, some experiments use a retro-reflected trapping

beam that interferes with itself in a standing wave, creating a trap with a strong

intensity gradient axially which results in a strong trap in 3 dimensions. However,

the atom populations spread out across nodes spaced λ/2 apart, instead of a single

potential minimum. We instead use a crossed beam configuration that has strong

confinement along all 3 directions in the center of the trap. The crossed beam trap

in a bowtie cavity can be configured to either have a standing wave potential or a

traveling wave in the trapping region. I discuss these configurations in section 1.3.2.

A more complete discussion of optical dipole trapping can be found in Ref. [10].

Evaporation and Feshbach Resonances

Feshbach Resonances The Feshbach resonance is a powerful tool that allows for

the manipulation of the interaction strength within ensembles of ultracold atoms,

using magnetic fields. Lithium is difficult to cool without using a Feshbach resonance,

but the resonance is abnormally large in lithium, resulting in a much more realizable

cooling process if the resonance is used [20]. The resonance is used to facilitate forced

evaporative cooling and to tune the chemical potential of an ensemble as the atoms

cool to degeneracy. In experiments that make use of metallic mounting schemes for

optical cavities, the strong magnetic fields that induce the resonant effect can be

distorted by the metallic mounts or induce forces on the mounting structure itself,

resulting in a disruption of the cavity. We have avoided these risks by using an

8


all-glass optical cavity.

To understand the basic idea behind the Feshbach resonance, assume that there

are two scattering channels for a pair of atoms in a scattering event, an energetically

allowed, or “open”, channel, and an energetically forbidden, or “closed”, channel. In

the classical regime the channels would not be coupled and the atoms would have

a scattering length corresponding to being localized in one potential. In actuality,

coupling between a closed and open channel can occur if the bound state of the closed

channel and the background level of the open channel are near the same energetic

level.

Take for example a pair of atoms in the triplet spin state approaching each other

for a scattering event. If the triplet spin state potential at large interatomic distances

is energetically close to a bound state of the singlet potential, a Feshbach resonance

occurs and the atomic scattering length changes based on the strength of the reso-

nance. A magnetic field can be used to shift the relative energy of the closed and

open channels to tune the scattering length. In figure 1.3 I have included a figure

that shows the open and closed channels during a scattering event.

Figure 1.3: Open and closed channels near a Feshbach resonance [5]

9

1.2 Optical Cavities Introduction

Forced Evaporative Cooling Evaporative cooling can be used to reduce the tem-

perature of an atomic ensemble by first trapping the atoms in a potential well and

then reducing the confinement of the trap adiabatically. As the trap’s confinement

is reduced, the most energetic atoms escape the trap and the atoms rethermalize

at lower temperature. In our experiment we will use a crossed beam dipole trap

in a cavity to confine the atoms and lower the optical power to decrease the trap

confinement.

We will make use of the Feshbach resonance of lithium to facilitate rethermaliza-

tion as atoms are released from the trap. If the rethermalization time is too long

compared to the vacuum-limited lifetime of the atomic ensemble, it is not possible

to increase the phase space density. If a Feshbach resonance is used, the scattering

length is tuned to cause the atoms to interact and rethermalize on faster time scales,

reducing opportunity for losses and maximizing phase space density.

1.2 Optical Cavities

1.2.1 Cavity Properties

In the simplest scenario, an optical cavity is a pair of mirrors facing each other [17].

Incident light is resonant with the cavity if the following condition is met:

d =qλ

2(1.8)

where d is the length of the cavity, λ is the wavelength of the light, and q is an integer

multiple known as the longitudinal mode order.

10


Figure 1.4: Linear cavity of length d

d

Mirror 1 Mirror 2

Incoming Beam Transmitted Beam

The separation between resonances in frequency space is known as the Free Spec-

tral Range (FSR), given by

νf =c

2d(1.9)

where c is the speed of light. The admitted light is reflected within the cavity until

transmitted through one of the mirrors (or scattered, or absorbed).

The cavity mirrors have inherent reflectivities, losses, and transmissions that de-

scribe how likely a photon will be lost, reflected, or transmitted through a mirror

interface. Cavity resonances have a finite linewidth, taken as the full width at half

max (FWHM). Finesse (F) is a ratio of the FSR (νf ) to the cavity linewidth (δνcav)

and is also a measure of the average number of reflections a photon will experience

within the cavity, such that

F =νfδνcav

(1.10)

We now need to determine the finesse in terms of the amplitude of the field

reflected from (r = Er/Ei ) and/or transmitted through (t = Et/Ei) the cavity

11


Figure 1.5: Diagram of ray tracing in a bowtie cavity

Reflected and Leakage Beam

Incoming Beam

Ec

Er

input mirror. The ring cavity we constructed is a 4 mirror symmetric bowtie (beams

cross in the center at right angles, see figure 1.5). If the 4 mirrors have identical

amplitude reflectivity, r, the relationship between the incident (Ei) intra-cavity (Ec),

and reflected (Er) field amplitudes is as follows:

Ec = tEi + tr4EceikL (1.11)

Er = tr3EceikL − rEi (1.12)

Notice that outside the cavity, the light trickling out of the cavity interferes with

the light reflected off the cavity mirror and that the cavity light picks up a phase

shift as it travels within the cavity. Solving for the power reflection and transmission

12


coefficients of the cavity input coupler as ratios of field amplitudes yields the following:

R =

∣∣∣∣∣ErEi∣∣∣∣∣2

=r2

1 + r8 − 2r4 cos(kL)(1.13)

T =

∣∣∣∣∣EtEi∣∣∣∣∣2

=t4

1 + r8 − 2r4 cos(kL)(1.14)

The maximum amount of input power will be coupled into the cavity when R is

minimum, i.e. when cos(kL) = −1, such that

C = 1−Rmin = 1− r2

(1 + r4)2(1.15)

In the limit that r → 1 the maximum achievable coupling with 4 identical mirrors is

75%. However, it is also possible to use mirrors of different reflectivity, and maximize

the coupling in a given configuration. This is referred to as “impedance matching” the

cavity. For all the work in this thesis, we assume mirrors with identical reflectivity.

The denominators of equations 1.13 and 1.14 define the linewidth of the cavity

through the coefficient of the cosine term. The denominators here are defined as

F which is known as the coefficient of finesse. The linewidth (νcav) of the cavity is

defined such that

kL

2= π

L

c(ν ± δνcav

2) = nπ ± 1√

F(1.16)

where n is some integer, and so combining we see that the finesse of a bowtie cavity

is

F =πr2

1− r4(1.17)

Extending the arguments above to the simpler configuration of a linear cavity, it can

13


be shown that the finesse of a linear cavity is

F =πr

1− r2(1.18)

The power buildup factor of a cavity is the ratio of intracavity power (Pc) to input

beam power (Pi), given by

PBF =PcPi

=1− r2

(1− r4)2(1.19)

for a bowtie cavity. Thusfar I have assumed that all light incident on a mirror is

either transmitted or reflected, such that

R + T = 1 (1.20)

In actuality, some power is scattered or lost on any optical surface, so the above

equation becomes

R + T + L = 1 (1.21)

where L represents losses. Losses effectively reduce the reflectivity of the cavity,

broadening the cavity linewidth and reducing power buildup, and reduce the coupling

efficiency, particularly in the limit where the transmission and losses are of similar

order.

1.2.2 Cavity Modes

Thus far I have treated optical fields as rays. In actuality they vary both longitudinally

and transversely. The profile of the electric field of a laser is actually a solution to

14


the paraxial Helmholtz equation, and the form of such solutions is given by

Emn(x, y, z) = E0w0

wHm

(√2x

w

)Hn

(√2y

w

)e−(x2+y2)(

1

w2+ik

2Rc

)−ikz−i(m+n+1)ζ(z)

(1.22)

where w is the spot size, w0 is the beam waist, Hn are the Hermite polynomials,

Rc is the radius of curvature, and ζ is the Guoy phase shift. The beam mode is

characterized as the TEMmn mode, where TEM stands for Transverse Electric Mode

and m and n refer to the indices of the Hermite polynomials. Most lasers have a

roughly Gaussian transverse intensity distribution, i.e. they are well approximated

by the TEM00 mode. Inside an optical cavity, the beam mode is set by the path length,

geometry, and mirror curvature. If the incident beam of the cavity is mode matched,

meaning that the incident beam has the same spot size and radius of curvature as

the TEM00 mode of the cavity at the boundary conditions of the cavity, then only

the TEM00 mode will be coupled. In practice, beams can be multimode, where they

are some superposition of Hermite Gauss modes.

Mode matching can be achieved by modifying the incident beam’s radius of cur-

vature and spot size with a set of lenses. I have included a detailed procedure for

mode matching in Appendix B. If mode matching is ignored or calculated improp-

erly, higher order modes are excited inside the cavity. Different spatial modes possess

slightly different frequencies, and so when scanning through cavity resonances they

will appear as separate resonances. When trying to lock to a resonance, you can

only lock to one TEM mode at a time, so proper mode matching ensures maximum

transmitted power efficiency.

15


Cavity modes shift by a frequency

δν = (n+m)νfζg2π

(1.23)

where ζg is the Gouy phase shift, a phase shift compared to a plane wave of the same

frequency that occurs when a Gaussian beam propagates axially [9], given by

ζg = arctan(z

zr) (1.24)

I have included figure 1.6 to show the distinction between mode splitting and free

spectral range in frequency space.

Figure 1.6: Mode spacing vs FSR

FSR

Transverse Mode Spacing

TEM00

TEM10

TEM11 TEM00

TEM10

TEM11

TEM21TEM21

Frequency

Power

If a cavity mode is asymmetric (if beam has different spacial dimensions) or astig-

matic (if the beam waist or curvature are different in different directions), the cavity

16


calculations, including mode frequency and mode matching conditions, must be done

separately using the height and width of the mode independently.

1.2.3 Cavity Matrices and q Parameter

At any point along the beam axis, the transverse spot size can be given by

w(z) = w0

√1 +

( zz0

)2

(1.25)

where z0 is the Rayleigh length, given by

z0 =πw2

0

λ(1.26)

The radius of curvature can be given by

Rc(z) = z(

1 +z2

0

z2

)(1.27)

Both the spot size and radius of curvature are encoded into the q parameter, given

by

1

q(z)=

1

Rc(z)+ i

λ

πw(z)2(1.28)

While the q parameter initially seems like an awkward construction, it can be

combined with ABCD matrix formulation to calculate the complex beam profile in

complicated systems. ABCD matrices are numerical constructs that represent how

17


beam profiles change as they interact with optical elements, according to

q2(z)

1

= k

A B

C D

q1(z)

1

(1.29)

where q1 and q2 represent the q parameters before and after the element respectively

and k is a normalization factor. ABCD matrices are well-known for many simple

optical elements and can be looked up in a textbook [17]. The above matrix equation

can be solved to give

q2(z) =Aq1(z) +B

Cq1(z) +D(1.30)

1.2.4 Cavity Locking

In an ideal world, lasers possess an infinitely narrow linewidth, never drift, and are

noiseless. In the same ideal world, cavities have a perfectly fixed path length that is

robust in the face of acoustic, mechanical, and thermal noise. Since the laboratory

exists in the real world, we instead use electronic feedback to correct for noise and

drift in both lasers and cavities [1].

General Locking Techniques At a very basic level, locking involves feeding in-

strument output to a locking device along with a desired setpoint that corrects any

deviation from the setpoint. The locking device calculates an error value by taking

the difference between the setpoint and the current output value and provides dy-

namic electronic feedback based on the error signal. As an example, consider a laser

injected into a linear optical cavity whose output is monitored by a photodiode. The

photodiode tracks the amplitude of cavity output and sends a voltage to a servo that

18

1.3 Experimental Design Constraints Introduction

provides control feedback to the laser, such that as the laser drifts away from the set-

point, the servo sends a signal to the laser controller that steers the laser frequency

back towards the cavity resonance frequency.

One problem with this scheme is that if the laser is locked to the slope of a

cavity resonance, due to the symmetry of the resonance a frequency shift across the

resonance could cause the lock to be lost as it the signal slope switches from positive

to negative, which would cause feedback in the wrong direction. Another issue is

that amplitude fluctuations in the laser could falsely indicate a frequency shift from

resonance. I discuss how we avoid these issues using the Pound Drever Hall method

in section 2.5.

1.3 Experimental Design Constraints

Power Constraints

Typical modern optical dipole traps use lasers that produce hundreds of watts of

optical power focused to a relatively small cross section to create a steep potential with

which to trap atoms. Lasers that produce this much power are potentially dangerous

and have relatively wide linewidths. They also cause thermal lensing effects on optics

leading up to the chamber and on the chamber windows themselves, whereby the high

intensity of the laser on the face of an optical element will impart enough thermal

energy to warp the face of the optic slightly, distorting the beam over time. By

contrast, using a less powerful but narrower-line laser in conjunction with a power

buildup cavity presents an elegant solution to these problems.

19


Magnetic Constraints

The rapid switching of high magnetic fields necessary to tune the Feshbach resonance

forced us to consider non-metallic mounting and construction options for the building

of the cavity. Even non-ferromagnetic alloys of steel are more ferromagnetic than

glass-type components, and the conductivity of metal allows for induction and eddy

currents in the neighborhood of the center of the chamber, distorting the magnetic

background. We therefore decided to attempt an “all glass” cavity, eschewing use of

metallic construction materials.

Vacuum Constraints

Keeping our experiment chamber as clean and evacuated as possible is a high priority

for our experiment. Our goal pressure for our main chamber is to sit at around

10−11 to 10−12 Torr, or roughly 15 orders of magnitude below atmospheric pressure.

Given our goal to make our cavity all glass, vacuum constraints have ruled out use

of conventional epoxies or resins, which out-gas too much to be used in experiments

pushing the limits of ultra-high vacuum [11].

1.3.1 Trapping Atoms in Cavities

Experimental trapping of atoms in cavities has been performed since the beginning

of the 21st century. In the following sections I outline some landmark experiments

demonstrating atomic trapping in cavities to give our work some context and to show

how our experiment differs from what has been done previously.

20


Linear Cavities

In 2001 Mosk et al. trapped spin mixtures of fermionic lithium atoms in a linear

standing wave optical dipole trap at λ = 1064 nm in a glass cell [14]. The cavity

mirrors themselves were placed outside of the cell and the cell windows were aligned

to the cavity mirrors at the Brewster angle to minimize intra-cavity losses. The ex-

periment took advantage of the Feshbach resonance of lithium to tune the interaction

length of the atoms in the mixture. Their linear cavity achieved a finesse of about

600 and a power buildup factor of 130. They achieved a 0.8mK ×kb depth trap that

held about 105 atoms after being supplied by a MOT with atom densities in the

1011 cm−3 range at ≤ 1mK. While they successfully trapped the atoms and avoided

issues inherent in using metallic mounting in an experiment using magnetic fields to

take advantage of a Feshbach resonance, they observed heating due to laser noise and

losses on the cell walls, even at the Brester angle.

In 2010 Bruno Zimmerman and the Tillman Esslinger group out of ETH Zurich

studied the microscopy of ultracold fermionic lithium while trapping atoms in a linear

standing wave resonator [19]. This time the mirrors were mounted inside the vacuum

chamber on metal pedestals. The cavity had a buildup factor of 6700 and a finesse

of 10200, allowing the dipole trap to have a maximum trap depth of 48 µK×kb at

the location of the MOT where the dipole beam waist was as large as 760 µm. The

cavity had very low losses resulting in high finesse and was able to successfully trap

atoms, but a metallic mounting scheme would be unsuitable for our experiment that

uses a Feshbach resonance. We would also like control over whether the trap was a

standing wave or traveling wave trap, which is not provided in a linear cavity.

21


Ring and Bowtie Cavities

In 2003 Kruse et al. trapped 85Rb atoms in a bidirectional standing wave trap in a 3-

mirror cavity [12]. They explored momentum transfer between atoms and the cavity

beams in both directions, noting that not only would photons redistribute themselves

into the opposite direction beam after scattering off of the Rb atoms, but they would

redistribute in between modes as well. Using p-polarized light they achieved a finesse

of 2500 and using s-polarized they achieved a finesse of 170000. The high-q cavity

produced 10W of intracavity power and had a trap depth of kb × 1.4mK using 799nm

light. They used the Pound Drever Hall locking technique to lock to cavity resonances

and were able to selectively couple to higher order modes and discussed the utility

of higher order modes when evaporatively cooling to degeneracy. They were able

to demonstrate 2-photon Raman transitions in a slowly moving standing wave trap,

where an atom would periodically scatter from one Raman beam to the other and

momentum would be imparted onto the atom. This cavity had high finesse using s-

polarized light, but due to the lack of a crossed beam configuration the atoms ended

up being stretched in the trap along the beam axis as opposed to localized at a point.

In 2011 Simon Bernon presented a high-finesse bowtie cavity for trapping and

cooling rubidium and for performing non-demolition measurements on ensemble spin

states [2]. The cavity mirrors were dual-coated for ultra-high reflectivity at 780 nm

and high reflectivity at 1560 nm. They achieved a finesse of 1788 for 1560 nm light.

Their mounting scheme was entirely metallic and adjustable either mechanically or

with piezo-actuators. They used their cavity to measure spin states of atomic ensem-

bles in a Mach-Zender interferometer, taking advantage of the high-Q capabilities of

their cavity to provide generate high signal-to-noise detection while minimally dis-

22


turbing the system. This system was similar to the one we would design, but using

Feshbach resonances for cooling Rb is not usually necessary, and so they made use of

metallic mounting and alignment schemes unavailable to us.

1.3.2 Atom-Cavity Physics

While trapping in a cavity does solve some of the problems mentioned in section

1.3 inherent when using a high optical power laser, dipole traps without a cavity are

actually cheaper and much easier to build. The trade-offs of using a lower power beam

in conjunction with the cavity might or might not be worth the effort of complicating

the overall experimental design. Another reason for using this scheme is that the

field of atom-cavity interactions involves a lot of interesting physics in its own right,

independent of the convenience of using the cavity for power buildup. By dual-coating

the mirrors of the cavity for both off-resonant trapping frequencies (1068nm, in the

IR) and on-resonant probing frequencies (671nm, in the red), the cavity can serve

double duty for trapping power buildup and atomically resonant probing. I have

outlined some examples of atom-cavity physics that would be possible to explore in

the future using the results of this thesis.

Cavity Cooling Techniques

One advantage offered by using a cavity-based dipole trap is that it has been shown

that you can use a slightly blue-detuned probe beam to reduce the kinetic energy

and temperature of trapped atoms. Atoms in the beam path of the cavity act as a

medium with a refractive index, changing the effective path length of the cavity. This

change in path length results in an upshift of the resonance frequency of the cavity.

23


However, if the cavity has a high finesse, the off resonant photons do not immediately

leak out of the cavity, resulting in a higher energy density in the beam. The system

energy is conserved through inelastic scattering events with the atoms, resulting in a

lowering of their kinetic energy and a drop in temperature. The cooling rate in some

experiments was increased by a factor of up to 5 compared to more conventional

means [16]. This method also has the advantage of being lossless, compared to the

more standard evaporative cooling techniques in which losses are an inherent property

of the system.

Quantum Non-Demolition Measurements

It is a basic tenet of quantum mechanics that performing a measurement on a sys-

tem will change that system. Quantum non-demolition measurements are designed

to indirectly measure the state of a system, resulting in a minimal impact on the

system being studied. Cavities have been used in conjunction with Mach-Zehnder

interferometers to make such measurements of the spin state of an atomic ensemble

[2]. As I have stated above, atoms in resonant light act as a dispersive medium, which

means they also induce a phase shift φ in the incoming light. If light makes N passes

through the atoms, it experiences a phase shift of Nφ. The phase shift depends on the

state of the atoms, so if we have a mixture of atoms in a combination of two states,

measuring the phase shift can give an indirect measurement of the atomic population

ratio given a known total number of atoms. Groups have used this phenomenon in

conjunction with high finesse cavities to create a large state-dependent phase shift

in the cavity light and compare the phase shifted light to that of a local oscillator

at the same frequency in a Mach-Zehnder interfermoter to measure the population

24


distribution of quantum gases in a non-destructive way [2].

Interference Control in Bowtie Cavities

Fine control of interference phenomena is inherent in a bowtie trap through manipu-

lation of the polarization of the incoming beams. If the incoming beam is polarized

in the plane of a cavity, at the center of the trap the crossed beams will have orthog-

onal polarizations, and no interference will be observed, creating a smooth trapping

potential over the entirety of the overlapping region. If the beam is polarized perpen-

dicular to the plane of the cavity, the polarization will be consistent throughout the

cavity, and the crossed beams will interfere with each other, creating a 1-D interfer-

ence pattern along the intersection of the beams. By adding a beam propagating in

the opposite direction of the original beam, in the case of in-plane polarization a 2-D

interference lattice pattern will emerge when the beams overlap in the center and a

1-D interference pattern will emerge in the case out-of-plane polarization.

Spontaneous Self-organization in an Optical Cavity

Spontaneous self-organization of atoms can occur in an optical cavity that possesses

an empty cavity mode [7]. The self-organization can also be used to cool and trap the

atoms further under the right conditions. Imagine an ensemble of atoms trapped in a

standing wave trap that are being pumped by near-resonant light. There is a phase-

position relationship in the trap, where a node is π out of phase with its neighboring

node. Assuming a smooth distribution of atoms, scattering from atoms at one position

in the standing wave are canceled out by the out-of-phase field contribution from its

neighbor, resulting in complete destructive interference in the scattering field.

25

1.4 Cavity Construction Techniques Introduction

A uniform atomic ensemble will still have density fluctuations, however, resulting

in a small scattered field with a random phase correlated to the density fluctuation.

The scattered field will create a high field-seeking potential for the atoms (assum-

ing that the field is red-detuned) which will tend to localize the initially uniformly

distributed atoms. The shifted atoms will resonantly scatter photons in a preferred

direction now in a superradiant mechanism similar to Bragg scattering, in which a

lattice of atoms will act as a diffraction grating along a specific direction and will

cause constructive interference of incident light along that axis. In the case of super-

radiance in a cavity, if the preferred scattering direction of the atomic ensemble is

into a cavity mode, the effect will be self-perpetuating and result in a coherent shift

of the atoms into potential wells corresponding to the newly populated cavity mode.

The coherent interaction between the atoms and the scattering field may result in

a damping effect on the atoms due to the shift and broadening of the cavity resonances

as the cavity field interacts with the atoms in a process similar to the one described

in the Cavity Cooling section above, where kinetic energy from the atoms is lost into

the shifting of the cavity modes. The damping results in atomic temperatures limited

by cavity linewidth instead of the Doppler limit which is present in more orthodox

atomic cooling mechanisms.

1.4 Cavity Construction Techniques

Optical cavities in vacuum are nothing new. An evacuated chamber provides a clean

and noise-free environment ideal for narrow linewidth cavities. Cavity construction

techniques in these environments vary based on constraints like space, cleanliness, and

magnetic induction susceptibility. In the planning stage of this project we surveyed

26


a variety of construction schemes which I will describe in the following sections.

1.4.1 Standard Optomechanics

The most straightforward way of putting a cavity in a vacuum chamber uses metallic

mirror mounts whose position and angle are governed by a set of either mechanical or

piezoelectric actuators. Piezo devices expand or contract when supplied an electric

voltage and allow for fine remote control of the mirror mounts. The leads of the piezo

devices can be run through the side of the chamber which allows for instrument control

outside the chamber [3]. The mirror mounts are bolted to a metal plate which is

secured inside the vacuum chamber. Metallic mounting parts can have very low levels

of outgassing. This scheme has the advantages of being highly vacuum compatible and

allows for experimental adjustment in situ via the piezo actuators. It was deemed

unsuitable for our experiment because the rapid switching of the Feschbach coils

would create eddy currents in all conductive metallic alloys, disrupting the magnetic

background uniformity in the experiment. The back electromotive force can also

induce instability in the cavity itself. Many metallic alloys used for optomechanics

are also ferromagnetic to at least a small degree, further distorting the background

magnetic field in the chamber.

1.4.2 Monolithic Techniques

Monolithic optomechanics sacrifice adjustability for the sake of avoiding problems

inherent in metallic mounting techniques. We surveyed a variety of possible bonding

options before settling on one.

27


Epoxy

Some experiments avoid using metal components by using epoxy resins to fix glass

components together to create a mounting scheme similar to the metallic one men-

tioned above. UHV-compatible epoxies are commercially available, but are only suit-

able in pressures 10−10 Torr or higher. Our experimental chamber must operate at

lower pressures, between 10−11 and 10−12 Torr. We were also concerned that whatever

outgassing occurred from the epoxies would increase absorption and scattering losses

and decrease the finesse of the cavity to an unacceptable degree.

Optical Contacting

Optical contacting is a phenomenon whereby two surfaces that are flat on molecular

scales and are extremely clean are bonded together purely by intermolecular forces

when pressed together. This method has the advantages of being ultimately clean,

involving only glass materials, and highly stable, but requires surfaces to be flat

and clean to a difficult-to-attain degree. A more serious problem inherent in optical

contacting is that the working time during the bonding is very low, and does not leave

room for error when aligning the cavity. Given that we were trying to affix circular

mirrors to flat mounts without obstructing or distorting any of the mirror faces, and

that we would have a very short working window during the contacting phase while

trying to align the cavity components, we deemed the method more trouble than it

was worth when compared to the next option.

28


Hydroxide Catalysis Bonding

We ended up using a bonding method recently used in several high-precision appli-

cations such as LIGO and the LISA Pathfinder mission [8]. In hydroxide catalysis

bonding, a small amount of bonding solution (sodium silicate in our experiment) is

pipetted onto the interface of two flat, clean glass surfaces. The hydroxide solution

dehydrates the silicate surface structures of the glass and causes them to bond to-

gether, leaving behind structurally rigid siloxane polymerized chains at the interface

that holds the surfaces together as the water evaporates. Silicate mixed in allows

for filling of gaps in the surfaces caused by surface mismatch or poor surface quality,

making this method far more forgiving than optical contacting. If the surfaces are

well matched, the bonding site is of optical quality.

29

Chapter 2

Development Process and Methods

2.1 Prior Work

Some first stage prototyping and testing was done by Sarah Khatry for this project.

She put together a table-mounted bowtie cavity using 1” high reflectivity mirrors in a

1068nm beam line, observed the cavity resonances, helped develop the mathematical

modeling code used for calculating beam parameters, and designed and assembled

some of the locking setup. I have included a picture of the cavity setup in figure 2.1.

2.2 Hydroxide Catalysis Bonding Tests

One of the first experimental goals of this project was to determine the adaptability of

hydroxide catalysis bonding (HCB) to UHV and high precision environments. To that

end I performed a variety of tests to validate HCB for use in our experiment. Much of

the currently existing literature suggests that a high degree of cleanliness, smoothness,

and flatness is required of the elements to be bonded successfully. However this

30

2.2 Hydroxide Catalysis Bonding TestsDevelopment Process and Methods

Figure 2.1: Preliminary Bowtie

literature is centered around space-based astronomical applications like the LISA

Pathfinder mission [8], and so bond strength in the face of being launched into space

has a higher priority in the literature than it does in our experiment. We needed to

determine if bonding curved and rough surfaces together was an option for our setup.

Preliminary Bonding Tests I bonded commercially available microscope slides

to each other to determine how difficult it would be to bond non-ideal surfaces. These

slides had been sitting in a dusty drawer for some time and were not ideal optical

elements. I cleaned the slides in distilled water and immersed them in an ultrasonic

bath of methanol for 20 minutes. I prepared a 4:1 by volume distilled water to sodium

silicate solution and partially immersed a closed beaker containing the solution in an

ultrasonic bath for 30 minutes. I then attempted to bond 4 pairs of slides to each

other using 0.5, 1, 1.5, and 2 µL of bonding solution and left the bonds to cure for 4

days, after which checking just for the presence of a bond between non-ideal elements.

All 4 samples bonded.

31


In the next test I bonded a curved, commercially fine-ground surface to a polished

flat one. I stood a 1” diameter optical-quality mirror on edge and slid 2 right angle

prisms underneath to fix the mirror in place and bonded along the lines of contact

between the mirror and prisms, as per the diagram in figure 2.2. I cleaned the mirror

Figure 2.2: Mirror Bonded to Right Angle Prisms

Mirror

Prism Prism

and prisms in distilled water and methanol and prepared a sample of bonding agent

as described above. I used 2 µL of solution along the interfaces between the mirror

and each of the prisms. After waiting to cure for 4 days, I determined once again

that the bonding was successful, though it could be pulled apart by hand.

Bond Strength To put a quantitative measure on how strong the bond was using

the hydroxide catalysis bonding procedure, I bonded smooth borosilicate glass spheres

0.5 cm in diameter to smooth glass microscope slides. Once the bonds were completed

and allowed to cure, to determine bond strength for a given bond I fixed a wire to

the top of the marble and hung a weight off of the other end of the wire, hanging off

the edge of a desk. I have included a diagram of the setup in figure 2.3 and tabulated

the resulting data in table 2.1.

32


Figure 2.3: Mirror Bonded to Right Angle Prisms

Weight

MarbleWire

Glass Slide on Desktop

Table 2.1: Bond Strength Test ResultsVolume (uL) Breaking Weight

for 4:1 Solution (N)Breaking Weightfor 3:1 Solution (N)

0.5 1.5 4

1 3 6

1.5 6 7

2 7 7

When testing the bond strength I hung progressively heavier weights off the edge

of the desk until the bond broke, incrementing by 50g weights. We were concerned

that using too much bonding agent in the experiment would result in water vapor

outgassing in the chamber, so we tested both bond strength for different volumes of

solution and bond strength for different solution concentrations.

We concluded that using 3:1 water to sodium silicate solution was more more

likely to limit the outgassing load and provide better bond strength overall.

33

2.3 Bowtie Cavity Design Development Process and Methods

Cure Timing In the literature, bond strength was deemed to be maximized if the

bond was left to cure for at least 4 days [8]. When we bonded two surfaces together

throughout experiments, we used alignment jigs that might lack adequate long-term

vibrational or mechanical stability, and so wanted to minimize the amount of time

the optic spent curing while attached to the mount in case someone bumped a table

or vibrations caused a bond to break. To that end, we wanted to ascertain how soon

it was safe to remove optics from the mounting jigs and allow them to bear their own

weight.

To do so, I set up a series of 1/2” mirrors bonded to microscope slides and allowed

them to cure over different time periods, including a half hour, an hour, 2 hours, 4

hours, 12 hours, 24 hours, and 2 days. After the allotted curing times I would

qualitatively determine the strength of the bond. I judged that after an hour or

two, the bond could withstand at least minimal incidental force and could probably

be removed from its mount safely. While this test was by no means quantitative,

it provided us with a reasonable procedure to follow through the later experimental

bonding stage, as we left the bonds to cure for 2 hours before removing the optics from

their mounts but still waited 4 days before moving or touching the bonded optical

assembly.

2.3 Bowtie Cavity Design

The bowtie cavity for use in the experiment must allow for certain design restrictions.

It must fit inside the main chamber while accommodating the MOT beams. The

mirrors must have a reflectivity and power buildup factor high enough to create a

deep dipole trap, without having such a narrow linewidth that vibrations prevent

34


robust locking of the laser to the cavity. It must also allow access to both a beam

reflection and transmission line through the chamber’s viewports. Figure 2.4 shows a

digital model of the entire vacuum chamber apparatus, including the 2D MOT cross,

3D MOT, and various pumps and gauges.

Figure 2.4: Complete Experimental System

I have also included a model of the inside of the experimental chamber, including

the cavity, MOT beams, viewports, and cavity beam lines. The red cylinders are

the MOT beams and the black lines inside the cavity are the trapping beams. To

accommodate the MOT beams, the cavity mirrors form a square 59 mm to a side.

In figures 2.6 and 2.7 I have included a model of the cavity itself. The mirrors

are mounted on pairs of pentaprisms with a 5mm clear aperture, shown in Fig. 2.6.

These 5-sided fused silica prisms happen to have an ideal height for mounting the

35


Figure 2.5: Experimental Chamber

cavity mirrors 10.5 mm above the ring-shaped Zerodur spacer, as shown in Fig. 2.7.

The whole assembly rests on four Viton rubber o-rings inside the chamber which help

isolate it from acoustic excitations. The reflected beam exits the chamber at an angle

that would cause it to clip on the viewports, so we have included in the design a right

angle prism bonded to the back of the cavity mirror to redirect the beam outside the

cavity so that it can be monitored.

Figure 2.6: Cavity Mirror Mounted on Pentaprisms

36

2.4 Modeling Development Process and Methods

Figure 2.7: Model of Monolithic Bowtie Cavity for Experiment

The mirrors themselves are high reflectivity and low loss. To improve coupling

efficiency into the chamber, the input mirror is at a slightly lower reflectivity at

r ≥ 99.98% compared to the r ≥ 99.992% reflectivity of the other mirrors. It is also

important to note that the mirrors themselves are relatively far from the viewport

windows, so mode matching schemes that require a lens within about 10 cm of the

input mirror are impractical.

2.4 Modeling

Given the above design for the experimental bowtie cavity, we numerically modeled

the dipole trap depth and shape following the arguments I made in 1.1.2. We also

modeled how the power buildup factor would change based on mirror reflectivity,

described in section 1.2.1.

Note that the power buildup is across all modes, so mode matching is necessary to

assure that all the power resides in the TEM00 mode we use for trapping. I included

37

2.5 Pound Drever Hall Locking Development Process and Methods

Figure 2.8: Power Buildup for Bowtie Cavity

a graph of power buildup instead of just a number based on our mirror specs because

the mirror specs are only given in terms of limits, so we do not have a precise quoted

value for mirror reflectivities. Also note that the basic calculation of the power buildup

factor does not take into account the input impedance of the cavity as described in ??,

so imperfect coupling could result in a lesser PBF. Assuming variable input power,

the trap depth is 3.23 µKW

of intracavity power and the harmonic trapping frequency

of the trap is 492×√P Hz where P is the intracavity power in Watts.The trap shape

and depth are shown in figures 2.9 and 2.10, for an assumed intracavity power of 300

W.

2.5 Pound Drever Hall Locking

In section 1.2.4, I mentioned that a couple of the problems with basic locking tech-

niques are that frequency and intensity fluctuations are indistinguishable and that

shifting off resonance produces a symmetric decrease in amplitude, meaning that you

38


Figure 2.9: Dipole potential at the center of the cavity, in the plane of the cavity

(m)

(m)

TrapDepth(uK)

can not determine what side of the resonance you are on from the amplitude alone.

The Pound Drever Hall locking method solves both above problems by looking at the

derivative of the cavity resonance, which is anti-symmetric about the center, instead

of the side of the resonance itself and by looking at the reflection off the cavity in-

stead of the transmission line [4]. To make use of the derivative of the resonance,

we put a sinusoidal frequency or phase modulation on the main line of the beam. If

the frequency of the laser has drifted above that of the resonance, the modulation

will be in phase with the intensity fluctuations resulting from moving in and out of

resonance. If we are below the resonance, the frequency modulation and intensity

fluctuations will be 1800 out of phase. We can provide electronic feedback by com-

paring the phases of the intensity and the modulation, determining which side of the

resonance we are on, and sending this information to a servo which provides feedback

to the laser controller.

39


Figure 2.10: Dipole potential at the center of the cavity, perpendicular to the planeof the cavity, cutting along one of the beams.

(m)

(m)

Trap Depth(uK)

To mathematically model the error function produced using the Pound Drever

Hall technique, let us assume that we have a lossless and symmetrical cavity. An

incoming beam in the form

E = E0eiωt (2.1)

will have some portion reflected. We define the ratio

F (ω) =ErefEinc

=reiω/FSR − 1

1− r2eiω/FSR(2.2)

as the ratio of the reflected and incoming beams’ electric fields. Let us now put a

phase modulation on the beam with modulation frequency Ω. The incoming beam

40


electric field becomes

Einc = E0ei(ωt+βsin(Ωt)) (2.3)

where β is the modulation depth. We can expand the above equation in terms of

Bessel functions, giving

Einc = E0[J0(β)eiωt + J1(β)ei(ω+Ω)t − J1(β)ei(ω−Ω)t] (2.4)

The above equation indicates that there are three beams to consider: one at the

carrier frequency ω and then sidebands at ω±Ω. Here we assume that all the power

is in the carrier frequency or the sidebands, and higher order modulations are ignored,

which is reasonable for β < 1. The carrier and sidebands have powers

Pc = J20 (β)P0 (2.5)

and

Ps = J21 (β)P0 (2.6)

respectively. The reflected beam is similarly given by

Einc = E0[J(β)0F (ω)eiωt + J1(β)F (ω + Ω)ei(ω+Ω)t − J1(β)F (ω − Ω)ei(ω−Ω)t] (2.7)

41


The power at the detector from the reflected beam is

Pref =(Constant terms)

+ 2√PcPsRe[F (ω)F ∗(ω + Ω)− F ∗(ω)F (ω − Ω)] cos(Ωt)

+ 2√PcPsIm[F (ω)F ∗(ω + Ω)− F ∗(ω)F (ω − Ω)] sin(Ωt)

+O(2Ω)

(2.8)

Here I have neglected constant terms and 2Ω terms because when we multiply the

signal at the detector with a local oscillator of frequency Ω using a frequency mixer,

terms with different frequencies will go to 0 and not contribute. If we assume a fast

modulation frequency and that we are close to resonance, the sin Ωt term dominates,

and mixing with a local oscillator supplying a sin(Ωt) signal extracts an error signal

of the form

ε = 2√PcPsIm[F (ω)F ∗(ω + Ω)− F ∗(ω)F (ω − Ω)] (2.9)

The shape of the error signal is shown in Fig. 2.11. The error signal possesses a

sharp, asymmetric curve centered at the resonance, providing a very convenient and

unambiguous signal to lock to.

In figure 2.5 I have included a circuit diagram to show how we implemented PDH

locking in our setup.

With the laser aligned to the cavity and scanning across cavity resonances, we

modulated the frequency of the laser with an electro-optic modulator (EOM) (Thor-

labs EO-PM-NR-C2) controlled with an amplified function generator signal. We used

a half waveplate to rotate the beam polarization to allow full transmittance through

the beamsplitter, then used a quarter waveplate to convert the beam polarization from

42


Figure 2.11: PDH Error Signal [4]

horizontal to circular. The beam reflected off the cavity is converted from circular to

vertical and reflected off the beam splitter into the photodetector. The photodetector

signal is high pass filtered to remove any DC offset and then mixed with the same

function generator signal (now called the local oscillator signal) that was running the

modulator. The mixer multiplies the signal from the cavity with the local oscillator.

The mixer outputs both a high frequency signal and a DC signal. We use a low

pass filter to remove the high frequency portion. The DC component contains the

phase-dependent differential signal derived above, which we then feed to a locking

servo. We adjust the servo setpoint to the zero of the differential signal.

Figure 2.5 shows an oscilloscope trace of the error signal going in to the locking

module of our laser, where the yellow trace is the cavity resonance on a photodiode

and the blue is the error signal.

43


Figure 2.12: PDH Locking Setup

44


Figure 2.13: PDH Oscilloscope Traces. The yellow trace is the cavity transmissionon a photodiode and the blue is the error signal generated from the beam reflectedoff the input coupling mirror.

45

Chapter 3

Results

Attempting to construct the experimental monolithic bowtie cavity using high quality

but expensive materials seemed unwise without first testing out the individual com-

ponents first. We wanted to ascertain whether or not hydroxide catalysis bonding a

functioning cavity together would work on any level, to assemble and test the locking

electronics setup, to check vacuum and bakeout compatibility of bonded cavities, to

build up experience mode matching to cavities, and to develop a procedure for pre-

cision alignment of a bowtie cavity without using traditional mounting schemes, all

before attempting to build the actual experimental cavity. The prototyping phase

proved useful in guiding the development of our techniques and in increasing our

likelihood of success when we actually attempt to build the experimental cavity.

46

3.1 Prototypes Results

3.1 Prototypes

3.1.1 Table-Mounted Linear Cavity 1

The first cavity we constructed simply consisted of 2 back-side polished mirrors coated

for high reflectivity at 780 nm, held in place by “D” mirror mounts. I manually

tested the mirror reflectivities by putting them in the laser line at normal incidence

and comparing the transmitted power to the total laser power using a photodiode.

The mirrors both had reflectivities of 99.17%. The measured reflectivities were low

compared to that of the mirrors intended for use in the main chamber, but this feature

was by design. The lower reflectivities allowed us to align the mirrors relatively quickly

to verify our electronic PDH locking setup and alignment techniques.

The setup was identical to that described in section 2.5, with the exception that we

modulated the beam frequency using a modulation input on the laser itself instead of

an EOM. A photodiode was placed in the path of the transmitted beam and another

in the path of the reflected beam coming off of a beamsplitter. A quarter waveplate

was placed in the path of the beam just before the cavity to rotate the polarization of

the reflected beam by a total of 900 so that the incident beam is transmitted through

the beamsplitter but the beam reflected off the cavity was fully reflected through

the same cube. The signal from photodiode measuring the transmitted beam was

displayed on an osilloscope and the signal from the reflected beam was mixed with a

local oscillator and used for feedback.

We placed an enclosure over the cavity with holes cut for the beams and floated

the optical table on compressed air actuators to minimize mechanical and accoustic

noise. To lock the laser to the cavity resonance, the resonance would have to have

47


drift in frequency space less than a linewidth away over short timescales. I measured

the short and medium term jitter and drift of the cavity respectively. On a scan-to-

scan time scale on the oscilloscope I measured that the resonance jittered by about a

sixth of the linewidth. The cavity mode center was given to drift by about the same

amount over a minute. While these measures were not particularly quantitative, they

provided evidence that the noise would not prevent us from locking the laser to the

cavity.

The laser was frequency modulated at 20 MHz and mixed with a local oscillator

at the same frequency, resulting in a differential signal at corresponding resonance

frequencies produced by the mixer. The differential signal was passed through a low-

pass filter and fed into a locking servo which provided feedback to the laser. Using

this scheme we successfully obtained a differential error signal and locked the laser to

the cavity.

3.1.2 Monolithic Linear Cavity 1

I used the same mirrors from the 780 nm table-mounted cavity to construct a mono-

lithic linear cavity prototype. I used a 2”x1”x1” borosilicate glass rectangular prism

as the mounting surface. I fixed the prism to a 5-axis stage mount in the line of the

780 nm laser used in the table-mounted setup. I arranged the mirrors in mounting

jigs fixed to the optical table using metal posts and right angle mounts. A picture

of the jig setup is shown in figure 3.1. One of the mirrors was mounted in a 2-axis

translation stage epoxied to a “D” mirror mount and the other was mounted on a

kinematic stage similarly epoxied to a “D” mirror mount. I aligned the mirrors on

the surface of the prism, relying on the kinematic and translation stages for align-

48


ment capabilities. I placed a photodiode in the transmitted beam line and aligned

the cavity using the stages until I observed what I deemed to be optimal transmission

resonances on an oscilloscope connected to the photodiode.

Figure 3.1: Jig Setup for Linear Monolithic Cavity 1

Once I deemed the cavity to be aligned, I used a micro manipulator tool to slide

3 mm right angle prisms bearing 1.5 µL sodium silicate solution into position under

the curved edges of the mirrors to fix the mirrors in place. I then bonded the bases

of the right angle prisms to the rectangular base prism and left the piece to cure for

several days.

Once the cavity was bonded and the bonds cured, I aligned it in the 780 nm

beam to determine whether or not the mirrors stayed aligned through the bonding

process and to set a benchmark performance for comparison when I put the cavity

in a vacuum chamber. The cavity stayed aligned enough to see strong resonances

comparable to those before the bonding. I measured the cavity linewidth and FSR

49


by putting 26 MHz frequency sidebands on the cavity line as a frequency reference

for the oscilloscope. By measuring the scope units corresponding to the sideband

separation, I could establish a conversion rate between scope timing and absolute

frequency while scanning.

I then measured both the medium and short-term noise on the cavity. Both the

medium and long term drift and jitter were in line with what we observed in the table

mounted linear cavity in that they shifted a fraction of a linewidth in each case as

long as the cavity was covered to minimize noise due to air currents and acoustics.

Once I had obtained reference performance data for the monolithic linear cavity, I

constructed a vacuum chamber shown in figure 3.2 out of standard stainless steel

vacuum parts.

Figure 3.2: Vacuum Chamber for Linear Monolithic Cavities

50


Table 3.1: Monolithic Linear Cavity 1Test FSR

(GHz)Linewidth(MHz)

Finesse Mode Spacing(MHz)

Post-Bonding 6.9 ±0.13 14.6 ±0.3 471 ±9 160 ±3In vacuum 6.9 ±0.13 35.6 ±0.7 193 ±4 166 ±3Post Bake 6.9 ±0.13 35.0±0.7 196 ±4 163 ±3

I sealed the cavity inside the vacuum chamber, aligned the 780 nm laser through

the cavity and vacuum chamber, and pumped the chamber down using a turbo pump

backed by a roughing pump. Once the cavity was pumped down to minimum pressure

at 5.8 × 10−9 Torr, I observed the FSR, linewidth, and mode spacing the same way

as mentioned above. I then wrapped the chamber in silicon heating strips and slowly

baked the cavity from room temperature to 1250 C over the course of 24 hours to

determine if the cavity could withstand the baking process we were likely to use in

our main experiment. Once the chamber had been at 125 0 C for about 12 hours,

I slowly decreased the temperature of the chamber back to room temperature and

retook cavity linewidth, FSR, and mode spacing measurements. All of the parameters

measured post-bonding, in vacuum, and post bakeout are given in Table 3.1.

3.1.3 Monolithic Linear Cavity 2

We bonded together a second linear cavity to build upon our results from the first.

Some key changes we made were that we used mirrors with higher reflectivity, mounted

the mirrors on top of dove prisms bonded to a BK7 rectangular prism, and increased

the length of the cavity to better model the longer length of the bowtie we were likely

to build for experiment. The higher mirror reflectivity and increased cavity length

increased the difficulty we faced in alignment and mode matching, simulating the

51


challenges we were likely to face when building the experimental bowtie. We also

ground flats onto the bottom edges of the mirrors in lieu of using right angle prisms

to hold them in place to determine the viability of a simpler mounting scheme. The

mirrors were 99.97% reflective to perpendicular incidence of 780 nm light. While the

cavity did eventually bond, bonding along the bottom flat of the mirror took a few

attempts and overall seemed less robust than the right angle prism method.

I used ray matrix formalism to numerically calculate mode matching conditions

for the cavity. I matched the collimated laser mode output from an optical fiber (350µ

m waist) to the cavity mode using a 150 mm lens and a 50 mm lens in a Galilean

telescope where the lenses were placed 200 mm apart and the 50 mm lens was placed

2 cm from the input cavity mirror. A more detailed description of mode matching

is included in Appendix B. Using this scheme I managed to get about 95% of the

intracavity power into the TEM00 mode, determined by sweeping through a full FSR

and measuring the amplitude of each resonance. I verified that the coupled mode was

TEM00 using a camera positioned on the other side of the cavity. An image of the

mode is shown in figure 3.3.The other mode with observable population was higher

order, shown in figure 3.4

Figure 3.3: TEM00 mode transmitted through the cavity

52


Figure 3.4: Higher Order Mode on Camera

Table 3.2: Monolithic Linear Cavity 2FSR(GHz)

Linewidth(MHz)

Finesse

1.82 ±0.03 5.7 ±0.1 316±6

This cavity had noticeably lower transmission throughput compared to the first

monolithic linear cavity. I measured the reflected beam with a photodetector and

observed that the TEM00 resonance dip was only 9% of the total reflected beam

amplitude. In a cavity with no losses, resonant and mode matched light should be

transmitted and completely interfere with the reflected light, so our 9% figure indi-

cates that our cavity had high losses from some source, potentially the mirror coating

which was not specified to have low loss compared to the transmission coefficient.

3.1.4 Table-Mounted Bowtie

As a precursor to attempting to bond a monolithic bowtie prototype we built a table-

mounted bowtie cavity. The mirror separations were similar to those to be used in

the experiment. The positions were set by using an aluminum plate with holes drilled

at the correct positions to fix the mirrors mounts to, as shown in Figure 3.5.

53


Figure 3.5: Table-Mounted Bowtie Cavity

Table 3.3: Table-Mounted BowtiePolarization FSR

(GHz)Linewidth(MHz)

Finesse

Horizontal 1.05 ±0.02 5.05 ±0.1 208 ±4

I used the paraxial ray matrix formalism to determine that a 100 mm and 50 mm

lens pair in a Galilean telescope would match the collimated 280 µm waist input beam

mode to that of the cavity (procedure described in Appendix B). After mode match-

ing, about 96% of the intracavity power was in the TEM00 mode. I measured the

FSR, linewidth, and finesse using a photodetector in a manner identical to how I have

above with one exception. I attempted to measure the parameters for both horizon-

tally and vertically polarized light. Cavity losses were so much greater with vertically

polarized light that there was not enough signal to noise to take measurements.

54


3.1.5 Table-Mounted Linear Cavity 2

I aligned two of the high reflectivity, low loss Layertec mirrors to be used in the

experimental bowtie in a linear cavity to determine their properties. We did not have

time to fully analyze this cavity, but being concerned with our apparently relatively

lossy mirrors, we wanted to test our low loss mirrors to verify that we were not

making some systematic mistake with our cavity construction techniques. The cavity

was aligned in a 1068 nm laser beam, frequency modulated at 1 MHz with an electro-

optical modulator. A photodiode measured the light transmitted through the cavity.

I immediately observed that while there was visible scatter off of the face of the

Thorlabs mirrors when viewed in the infrared, when the Layertec mirrors were illumi-

nated with 500 mW of 1068 mn light and observed through an infrared viewer there

was no detectable scatter. This was a good indicator of their low loss characteristics

and cleanliness. I then measured the linewidth and FSR, and calculated the finesse

of the cavity, as shown in Table 3.4.

Table 3.4: Table Mounted Linear Cavity 2FSR(GHz)

Linewidth(kHz)

Finesse

1.01±0.02 250 ±5 4000 ±24

Using equation 1.18, I determined that with a measured of finesse of 4000, our

effective reflectivity for the cavity mirrors was 99.961%. The quoted reflectivity was

99.97%. We found this result encouraging in that even if our prototyping mirrors

were not ideal, our mirrors for use in the experiment would be.

55

3.2 Monolithic Bowtie Cavity Results

3.2 Monolithic Bowtie Cavity

The final proof of principle for the experimental bowtie cavity was a monolithic cavity

that matched the design parameters and restrictions in our experiment. The proto-

type was a square cavity 59 cm on a side. We bonded the four Thorlabs mirrors from

the table-mounted bowtie on top of pentaprism pair supports that were bonded to

a borosilicate glass substrate. I have included pictures of the resulting cavity and of

the pentaprism support scheme in figures 3.6 and 3.7. The pentaprisms both serve

to more securely bond the mirrors than the flat substrate surface would alone and to

put the mirrors at the height planned for the experimental cavity in the experimental

chamber.

Figure 3.6: Monolithic Bowtie Cavity

56


Figure 3.7: Mirror Supported by Pentaprisms

3.2.1 Jig Design

To make sure that the diagonal arms of the cavity overlapped in the center we paid

particularly close attention to making sure there was no “twist” of the cavity mirrors

our of the alignment plane, such that cavity modes were centered on each mirror and

the mirrors were all in the same plane. To obtain precise positioning and angular

fidelity in the construction process, we designed and had machined a metal jig that

would set mirror locations and angles within narrow tolerances. I have shown a design

and picture of the jig in figures 3.8 and 3.9.

The mirrors are fixed in “D” mirror mounts and placed into grooves in the jig.

The inner entrance to each of the four grooves corresponds to the exact placement

57


Figure 3.8: Bowtie Cavity Alignment Jig Schematic

and angle of the mirrors in the cavity design. The mirrors are pressed against a

flat precision gauge acting as a stop at the designed location and angle. The jig is

screwed into a vertical translation stage to control the height of the cavity. We also

had built a removable cylindrical insert piece for the jig with 280 µm diameter holes

along the cavity beam line to act as a reference or guide during alignment.The jig

was designed to be of similar height to our substrate so that once the mirrors were

properly aligned in the jig, the stage could be lowered and the jig swapped out for the

substrate. Shown in figure 3.10 is a rendering of the jig with the insert and options

for fiber optic and pellicle alignment tools we decided not to use. The stage would

then be raised again to appropriate height to accommodate the pentaprisms in the

bonding process.

58


Figure 3.9: Bowtie Cavity Alignment Jig Model

Figure 3.10: Bowtie Jig with Insert on Vertical Translation Stage

The jig itself allowed for initial alignment, but for fine alignment we set up mounts

that allowed for vertical and horizontal tilt and for vertical and horizontal translation

to maximize our control of the mirror locations. The mounting scheme used a Thor-

labs Polaris mirror mount epoxied to the “D” mirror mount for tilt control and two

translation stages fixed to the Polaris mount for translational control. The mounts

were supported by 1” diameter steel rods fixed to the optical table.

59


3.2.2 Alignment Procedure

We aligned a 780 nm beam to the jig setup prior to inserting the mirrors into the

jig. After the beam was roughly aligned to the first arm of the grooved guides in the

cylindrical insert, we placed the mirrors in “D” mirror mounts into the jig and placed

a photodiode in the transmitted beam line outside of the cavity. Due to refraction, the

beam line shifted as it was transmitted through the first cavity mirror, so we adjusted

to realign the beam to the guide. Once the mirrors were all in place and referenced

to the jig using the precision gauge as a stop, we brought the mounting arms into

contact with the “D” mirror mounts and epoxied the mounts to the mounting arms.

I have included an image of the setup in Figure 3.11.

Figure 3.11: Mounting Setup (Top View)

60


The rest of the alignment process was based on selectively removing degrees of

freedom from the setup to ensure there was no unintentional walk-off of the beam in

the cavity that would result in cavity twisting. Once the beam line was aligned to the

first arm of the guide insert, we no longer touched the input coupling mirrors. We

then progressively aligned the mirrors in the cavity along the beam path to ensure the

beam passed through all arms of the guide insert. When the alignment was sufficiently

close, we began to see cavity resonances on the photodiode. We maximized the cavity

resonance signal by making slight changes to the cavity mirror alignment while making

sure the beam path adhered to the path dictated by the guide insert. I have included

a picture of the setup in the alignment jig in figure 3.12.

Figure 3.12: Mirrors Aligned with Jig

61


Once the mirrors were aligned properly, we lowered the stage with the jig at-

tached, leaving the mirrors situated in free space. We detached the jig and raised the

borosilicate glass spacer on the vertical translation stage up to the height needed to

accommodate the pentaprisms. We slid the pentaprisms into position under align-

ment and bonded them to the mirrors and the substrate using 1.5 µL of 3:1 sodium

silicate solution at each bond site. I have included a picture of the setup at this point

in figure 3.13. We left the bonds to cure for roughly 2 hours and then detached the

mirrors from the mounts and left to cure further for 48 hours.

Figure 3.13: Bowtie During Bonding Process

62


3.2.3 Verification of the Cavity Alignment

I used a 20 MHz modulation signal to measure the FSR and linewidth and calculated

the finesse of the cavity. Once again, I measured the parameters for horizontally

polarized light, as the vertically polarized light was entirely lost within the cavity.

I took the measurements immediately after construction and then a week later to

determine the effect of leaving the clean mirrors out in open air. I tabulated the

results in table 3.5 It is clear that leaving the mirrors out in open air when making

the experimental cavity will present unacceptable losses and line broadening.

Table 3.5: Monolithic Bowtie PrototypeTime FSR (GHz) Linewidth

(MHz)Finesse

Right after construction 1.075 ±0.02 2.35 ±0.04 437 ±9One week after construction 1.075 ±0.02 5.37 ±0.11 200 ±4

I also used a camera to take images of the transmitted beam outside the camera.

We observed about 90% of the power was in the TEM00 mode. I have shown images

of the modes resonating within the cavity in figures 3.14 and 3.15.

Figure 3.14: TEM00 mode of the monolithic bowtie cavity

63


Figure 3.15: Higher order mode of the monolithic bowtie cavity

We also checked the cavity twist after bonding to determine the vertical overlap

of the two diagonal beams in the cavity. I suspended a sharp (etched tungsten STM)

tip in a mechanical 3-axis translation stage and lowered the tip into the cavity beam

line. I lowered the tip into one arm of the beam line within 0.5 mm of the point

where the beams crossed until I saw a 50% drop in the cavity resonance amplitude.

I then translated the tip horizontally out of that beam line arm and into the other

and adjusted the vertical alignment until I saw a 50% reduction in the resonance

amplitude. I took the difference in vertical positions corresponding to the 50% dropoff

as the difference in height of the beams. This technique indicated that one beam was

20±16 µm higher than the other. This figure includes is the uncorrelated combination

of all random and systematic errors, including measurement uncertainty of the stage

postion, worst-case estimates of non-orthogonality of the translation stages, wedge in

the substrate, and variation in thickness of the viton O-rings. Given that the beams

are approximately 300 µm in diameter, we judged that the level of twist in the cavity

was acceptable for the purpose of trapping.

64


Estimation of Mirror Losses

While using these mirrors (that were never quoted to be low loss, only highly re-

flective) we had noticed that we consistently saw relatively low throughput in our

cavities, regardless of mode matching or geometry. Using equation 1.17, I determined

that, given a finesse of around 200 like we see in all of our monolithic cavities after

being out in air for any amount of time, the effective reflectivity of our cavity mirrors

is 99.2%. Comparing to the 0.03% measured transmission, we can approximate the

scattering coefficient to be about 0.77%. Most of our light that is not reflected is

being lost on the cavity mirrors, resulting in low throughput and wider resonances.

65

Chapter 4

Conclusions

We have seen promising results moving towards building the experimental cavity. We

have verified that the hydroxide catalysis bonded cavities can withstand moderate

bakeout temperatures and can sit in a vacuum chamber without large amounts of

outgassing. We have also developed a set of alignment procedures and jigs for aligning

bowtie cavities in free space and bonding them to a flat substrate.

While we do not yet have the experimental monolithic bowtie cavity constructed,

that step seems just around the corner. We chose not to construct it at the last

minute to avoid user error due to rushing and to perform further analysis on our ex-

isting cavities. The experimental cavity will be introduced into an ultra-clean system

where in-situ adjustment or correction is impossible, and risks of contamination are

unacceptable if they will disrupt the rest of the experiment. We need to get it right

the first time, and so we chose to play it safe.

To that end, some of the next few steps will include vacuum testing the prototype

bowtie and bringing it down to pressures we are likely to see in experiment, i.e. below

10−11 Torr. With the introduction of more bonding material in the bowtie compared

66

Conclusions Conclusions

to the linear due to more bonds being used, it is possible the bowtie will introduce

either more water vapor or more room contaminants or both into the chamber. A

bakeout stress test to push the limits of the maximum temperature and differential

expansion due to temperature the bonds can withstand might also be a good idea.

While most of the monolithic bowtie alignment process seemed to work as planned,

during alignment I noticed that the widths of the grooves used as beam guides in

the alignment jig insert seemed overly forgiving. We were initially concerned that

narrower guides might result in clipping that brought high losses, but we did not

observe any such phenomenon, and reducing the diameter of the alingment grooves

might give us tighter tolerances on cavity twist. We could also slot in a pellicle into the

alignment jig, which is a thin screen that shows beam lines that might indicate degree

of beam overlap inside the cavity during the alignment process. The bowtie cavity

with the Thorlabs mirrors did not provide high enough intracavity power to observe

spots on the pellicle, but the power will be much higher in the low loss cavity we will

build and might allow for its use. Additionally, future efforts to measure the vertical

offset of the cavity beams should try to remove possible sources systematic error, for

example by establishing tighter bounds on non-orthogonality and mechanical runout,

and by using a four-point measurement.

67

Appendix A

Part Numbers

68

Part Numbers Part Numbers

Table A.1: Part NumbersPDH Locking Equipment

1 MHz EOM Thorlabs E0-PM-NR-C2Locking Servo New Focus LB1005Mixer MiniCircuits ZAD-8+Input Amplifier MiniCircuits ZX60-100VHXLaser Piezo Controller Thorlabs MDT6984BEOM Amplifier Thorlabs HVA200Transmitted Beam Photodiode Thorlabs PDA10CReflected Beam Photodiode Thorlabs PDA10CS

Lasers780 nm Laser New Focus Velocity 6312780 nm Laser (Higher Power) Toptica DLX 1101068 nm Laser Koheras Boostik Fiber Laser

69

Appendix B

mode matching Procedure

To modematch an incoming beam to a cavity, the problem can be reframed to be to

match the incoming beam’s Rayleigh Length to that of the cavity mode’s and then

to position the beam optics so that the location of the beam waist of the input beam

coincides with position of the beam waist of the cavity mode inside the cavity.

For the sake of simplicity, I’ll show the procedure only for mode matching to a

linear cavity, but mode matching to a bowtie cavity uses a similar procedure that

involves more ray matrices due to there being more cavity mirrors. I describe the

theoretical underpinnings referenced in this section in 1.2.3.

The first step is to determine the beam modes both outside and inside the cavity.

The beam mode inside the cavity can be modeled using ray matrices, where we

assume an incoming beam travels one cavity length inside the cavity, reflects off a

cavity mirror, then travels the same distance inside the cavity and reflects off of the

other cavity mirror. The ray matrix representation of a beam traveling through space

is given by

70

mode matching Procedure mode matching Procedure

S =

1 L

0 1

(B.1)

and for reflecting off of a mirror with radius of curvature R the representation is

M =

1 0

−2/R 1

(B.2)

and so the ray matrix for a cavity is the product of

CAV =

1 0

−2/R 1

1 L

0 1

1 0

−2/R 1

1 L

0 1

(B.3)

Recall that the q parameter evolves with under a ray matrix under the transformation

q2(z) =Aq1(z) +B

Cq1(z) +D(B.4)

For the cavity to be stable, rays must trace back onto themselves on return trips,

so we can impose on our system that for return trips, q2 = q1. Solving the above

equation under this condition gives an equation for the q parameter in terms of the

cavity ray matrix ABCD parameters.

Cq2 + (D − A)q −B = 0 (B.5)

Solving the above equation for q gives us the q parameter for the cavity.

Outside the cavity, the q parameter can simply be measured using a camera to

71

mode matching Procedure mode matching Procedure

measure the beam1

e2radius at points along the beam line. Finding the location and

size of the beam waist outside the camera fixes the q parameter, such that if the

Rayleigh Length is

z0 =πw2

0

λ(B.6)

then the q parameter is

q(z) = z + iz0 (B.7)

where z is the position from the waist along the beam line. Assuming now that we

start at some position z along the beam line, using a lens pair we are able to match

the Rayleigh Length of the incoming beam with that of the cavity. The ray transfer

matrix for a pair of lenses with focal lengths f1 and f2 separated by a distance d apart

at a distance z from the initial beam waist is

X =

1 0

−1/f2 1

1 L

0 1

1 0

−1/f1 1

1 L

0 1

(B.8)

By solving for the imaginary part of the new q parameter and picking the lens focal

lengths and spacings correctly, you can match the waist size to that of the cavity

waist size. The real part of the new q parameter is the distance between the second

lens and the new waist. All that is needed to finish mode matching at this point is to

shift relative positions of the external beam lenses and the cavity until their waists

overlap.

72

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75

Acknowledgements

I would like to thank my advisor Kevin Wright for being an endless font of knowledge

of all things experimental physics and for guiding me through my time in graduate

school. I’d also like to thank Sarah Khatry for her prior work on this project and

Yanping Cai and Dan Allman for editing and being excellent labmates. I would also

like to thank Dwayne Adams for his excellent machining work on this project.

Last and certainly not least I’d like to thank my mom, dad, and sister (hi Jay)

for being loving and supportive.

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construction of monolithic all-glass optical...

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